Carlson-Levin Constant

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

Assume that $f$ is a Nonnegative Real function on $[0,\infty)$ and that the two integrals

$$\int_0^\infty x^{p-1-\lambda} [f(x)]^p\,dx$$ (1)

$$\int_0^\infty x^{q-1+\mu}[f(x)]^q\,dx$$ (2)

exist and are Finite. If $p=q=2$ and $\lambda=\mu=1$, Carlson (1934) determined

$$\int_0^\infty f(x)\,dx\leq \sqrt{\pi}\left({\int_0^\infty [f... ...right)^{1/4}\left({\int_0^\infty x^2[f(x)]^2\,dx}\right)^{1/4}$$ (3)

and showed that $\sqrt{\pi}$ is the best constant (in the sense that counterexamples can be constructed for any stricter Inequality which uses a smaller constant). For the general case

$$\int_0^\infty f(x)\,dx\leq C\left({\int_0^\infty x^{p-1-\lam... ...ight)^s\left({\int_0^\infty x^{q-1+\mu}[f(x)]^q\,dx}\right)^t,$$ (4)

and Levin (1948) showed that the best constant

$$C={1\over(ps)^s(qt)^t} \left[{\Gamma\left({s\over\alpha}\rig... ...lambda+\mu)\Gamma\left({s+t\over\alpha}\right)}\right]^\alpha,$$ (5)

where

$$s \equiv {\mu\over p\mu+q\lambda}$$ (6)

$$t \equiv {\lambda\over p\mu+q\lambda}$$ (7)

$$\alpha \equiv 1-s-t$$ (8)

and $\Gamma(z)$ is the Gamma Function.

References

Beckenbach, E. F.; and Bellman, R. Inequalities. New York: Springer-Verlag, 1983.

Boas, R. P. Jr. Review of Levin, V. I. ``Exact Constants in Inequalities of the Carlson Type.'' Math. Rev. 9, 415, 1948.

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/crlslvn/crlslvn.html

Levin, V. I. ``Exact Constants in Inequalities of the Carlson Type.'' Doklady Akad. Nauk. SSSR (N. S.) 59, 635-638, 1948. English review in Boas (1948).

Mitrinovic, D. S.; Pecaric, J. E.; and Fink, A. M. Inequalities Involving Functions and Their Integrals and Derivatives. Kluwer, 1991.